Computational Fractional Dynamical Systems
by Rajarama M. Jena, Subrat K. Jena, Snehashish Chakraverty
2Recent Trends in Fractional Dynamical Models and Mathematical Methods
2.1 Introduction
Mathematicians like Leibniz, L'Hôpital, Abel, Liouville, Riemann, and others conceptualized fractional calculus as the theory of integrals and derivatives of arbitrary real (and complex) order. Nonlocality, an intrinsic property of many complex systems, makes fractional derivatives suitable for modeling phenomena in various sciences and engineering disciplines. Fractional derivatives deal with the global evolution of the system rather than just focusing on the local dynamics. Therefore, fractional derivatives provide better representations of real‐world behavior than ordinary derivatives. Although fractional calculus is three centuries old, now it is very popular with scientists and engineers. The beauty of this subject is that fractional derivatives (and integrals) are not a local (or point) property. This subject, therefore, considers the history and nonlocal distribution effects.
2.2 Fractional Calculus: A Generalization of Integer‐Order Calculus
Consider an integer n, and when we say xn we immediately visualize multiplying x by itself n times to give the result. We still obtain the result if n is not an integer, but it will not be easy to visualize how. Further, 2e and 2π are hard to visualize, but they exists. Similarly, although the fractional derivative
is hard to visualize, it exists. ...